Problem with Bevel modifier
Thanks to hint from 4-K, I indeed found out that the problem was because of duplicates. To check it, I just moved single vertices in edit mode.

To remove them I did the following:


Open the edit mode for vertices.
Select all the vertices using A key.
Open special menu using W key.
Click "Remove Doubles".


Then Blender said that there were 8 redundant vertices, and I got correctly working bevel modifier.